General Relativity/Christoffel symbols

( << Back to General Relativity) Definition of Christoffel Symbols Consider an arbitrary contravariant vector field defined all over a Lorentzian manifold, and take at , and at a neighbouring point, the vector is at . Next parallel transport from to , and suppose the change in the vector is . Define: The components of must have a linear dependence on the components of . Define Christoffel symbols : Note that these Christoffel symbols are: <li>dependent on the coordinate system (hence they are NOT tensors) <li>functions of the coordinates Now consider arbitrary contravariant and covariant vectors and respectively. Since is a scalar, , one arrives at: Connection Between Covariant And Regular Derivatives From above, one can obtain the relations between covariant derivatives and regular derivatives: Analogously, for tensors: Calculation of Christoffel Symbols From , one can conclude that . However, since is a tensor, its covariant derivative can be expressed in terms of regular partial derivatives and Christoffel symbols: Rewriting the expression above, and then performing permutation on i, k and l: Adding up the three expressions above, one arrives at (using the notation ): Multiplying both sides by : Hence if the metric is known, the Christoffel symbols can be calculated.